Here is a development of the previous proof, which takes into account the special case when the circle in the z-plane passes through the origin and is transformed into a straight line (circle with infinite radius as Penrose calls it).
I noticed a small typing error in the attached .pdf file which I have now corrected.

"Equation (5) is of the same form as equation (1)" should be "Equation (5) is of the same form as equation (2)"


Vasco